\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx \]
Optimal antiderivative \[ \frac {4 b \,B^{2} d \,n^{2} \left (d x +c \right )}{\left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )}-\frac {b^{2} B^{2} n^{2} \left (d x +c \right )^{2}}{4 \left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )^{2}}+\frac {4 b B d n \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )}-\frac {b^{2} B n \left (d x +c \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{2 \left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )^{2}}+\frac {2 b d \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )}-\frac {b^{2} \left (d x +c \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{2 \left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )^{2}}+\frac {d^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{3}}{3 B \left (-a d +b c \right )^{3} g^{3} i n} \]
command
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{4} \, {\left (-\frac {2 i \, {\left (d x + c\right )}^{2} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )}^{2} g^{3}} + \frac {2 \, {\left (-i \, B^{2} n^{2} - 2 i \, A B n - 2 i \, B^{2} n\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{2} g^{3}} + \frac {{\left (-i \, B^{2} n^{2} - 2 i \, A B n - 2 i \, B^{2} n - 2 i \, A^{2} - 4 i \, A B - 2 i \, B^{2}\right )} {\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________